A circle in the first quadrant of the standard (x,y) coordinate plane is tangent to the x-… — ACT

A circle in the first quadrant of the standard $(x,y)$ coordinate plane is tangent to the $x$ -axis and is also tangent to the line $y = \frac{4}{3}x$ . If the center of the circle lies on the line with equation $y = 3x - 10$ , what is the radius of the circle?

Answer: 2

Answer

The radius of the circle is 2.

The correct answer is obtained by determining that the center of a circle tangent to the

x

-axis in the first quadrant has the form

(h, R)

where

R

is the radius. Using the distance from this point to the line

4x - 3y = 0

gives

h = 2R

to ensure the center remains in the first quadrant. Substituting

(2R, R)

into the line

y = 3x - 10

yields

R = 3(2R) - 10

, which solves to

R = 2

Step-by-Step Solution

Determine the relation between the circle's center coordinates and its radius.

The center is

(h, R)

where

R

is the radius.

Because the circle is tangent to the

x

-axis and lies in the first quadrant, the y-coordinate of its center must equal its radius.

Apply the point-to-line distance formula from the center to the line

y = \frac{4}{3}x

The relation is

|4h - 3R| = 5R

The distance from the center

(h, R)

to the line

4x - 3y = 0

must equal the radius

R

Solve the absolute value equation for

h

in terms of

R

Since

h > 0

, we find

h = 2R

The positive case

4h - 3R = 5R

gives

h = 2R

, whereas the negative case

4h - 3R = -5R

gives a negative

h

which violates the first-quadrant condition.

Substitute the center coordinates

(2R, R)

into the line

y = 3x - 10

and solve for

R

R = 2

The center lies on this line, so its coordinates must satisfy the line's equation.

Key Concept

Equations and Graphs of Circles

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic